scholarly journals Extremal lifetimes of persistent cycles

Extremes ◽  
2021 ◽  
Author(s):  
Nicolas Chenavier ◽  
Christian Hirsch
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
General Feature ◽  
Extreme Values ◽  
Persistent Homology ◽  
Poisson Approximation ◽  
Continuum Percolation ◽  
Coexistence Region ◽  
Death Time ◽  
Topological Features

AbstractPersistent homology captures the appearances and disappearances of topological features such as loops and cavities when growing disks centered at a Poisson point process. We study extreme values for the lifetimes of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation and the coexistence region. First, we describe the scaling of the minimal lifetimes for general feature dimensions, and of the maximal lifetimes for cavities in the Čech filtration. Then, we proceed to a more refined analysis and establish Poisson approximation for large lifetimes of cavities and for small lifetimes of loops. Finally, we also study the scaling of minimal lifetimes in the Vietoris-Rips setting and point to a surprising difference to the Čech filtration.

Bounded-hop percolation and wireless communication

2016 ◽  
Vol 53 (3) ◽  
pp. 833-845 ◽  
Author(s):  
Christian Hirsch
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Base Station ◽  
Telecommunication Networks ◽  
Base Stations ◽  
Continuum Percolation ◽  
Supercritical Regime ◽  
Close Relationship ◽  
Wireless Telecommunication ◽  
Minimum Number

AbstractMotivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process, we investigate the distribution of the minimum number of hops that are needed to reach some point of the base station process. In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the base station process as its intensity tends to 0. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the base station process in a given number of hops.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (4) ◽  
pp. 732-739 ◽  
Author(s):  
Bruce M. Brown ◽  
Sidney I. Resnick
Keyword(s):  
Stochastic Processes ◽  
Point Process ◽  
Time Scales ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extreme Values ◽  
Weak Limit ◽  
One Dimensional ◽  
Limit Process ◽  
Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (04) ◽  
pp. 732-739 ◽  
Author(s):  
Bruce M. Brown ◽  
Sidney I. Resnick
Keyword(s):  
Stochastic Processes ◽  
Point Process ◽  
Time Scales ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extreme Values ◽  
Weak Limit ◽  
One Dimensional ◽  
Limit Process ◽  
Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

scholarly journals Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

2013 ◽  
Vol 45 (1) ◽  
pp. 20-36 ◽  
Author(s):  
C. Hirsch ◽  
D. Neuhäuser ◽  
V. Schmidt
Keyword(s):  
Point Process ◽  
Open Problem ◽  
Poisson Point Process ◽  
Point Processes ◽  
Geometric Graphs ◽  
Continuum Percolation ◽  
Random Geometric Graphs ◽  
Spanning Forests ◽  
Percolation Thresholds ◽  
Relative Neighborhood Graph

The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point process X ⊂ ℝd is an open problem for dimension d>2. We introduce a descending family of graphs (Gn)n≥2 that can be seen as approximations to the MSF in the sense that MSF(X)=∩n=2∞Gn(X). For n=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity of Gn(X) holds for all n≥2, all dimensions d≥2, and also point processes X more general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surely X does not admit generalized descending chains.

scholarly journals Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

2013 ◽  
Vol 45 (01) ◽  
pp. 20-36 ◽  
Author(s):  
C. Hirsch ◽  
D. Neuhäuser ◽  
V. Schmidt
Keyword(s):  
Point Process ◽  
Open Problem ◽  
Poisson Point Process ◽  
Point Processes ◽  
Geometric Graphs ◽  
Continuum Percolation ◽  
Random Geometric Graphs ◽  
Spanning Forests ◽  
Percolation Thresholds ◽  
Relative Neighborhood Graph

The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point processX⊂ ℝdis an open problem for dimensiond>2. We introduce a descending family of graphs (Gn)n≥2 that can be seen as approximations to the MSF in the sense that MSF(X)=∩n=2∞Gn(X). Forn=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity ofGn(X) holds for alln≥2, all dimensionsd≥2, and also point processesXmore general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surelyXdoes not admit generalized descending chains.

scholarly journals Percolation in the signal to interference ratio graph

2006 ◽  
Vol 43 (2) ◽  
pp. 552-562 ◽  
Author(s):  
Olivier Dousse ◽  
Massimo Franceschetti ◽  
Nicolas Macris ◽  
Ronald Meester ◽  
Patrick Thiran
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Euclidean Distance ◽  
Critical Density ◽  
Continuum Percolation ◽  
Two Dimensional ◽  
Radio Communications ◽  
Communications Networks ◽  
Infinite Range ◽  
Independent Model

Continuum percolation models in which pairs of points of a two-dimensional Poisson point process are connected if they are within some range of each other have been extensively studied. This paper considers a variation in which a connection between two points depends not only on their Euclidean distance, but also on the positions of all other points of the point process. This model has been recently proposed to model interference in radio communications networks. Our main result shows that, despite the infinite-range dependencies, percolation occurs in the model when the density λ of the Poisson point process is greater than the critical density value λc of the independent model, provided that interference from other nodes can be sufficiently reduced (without vanishing).

scholarly journals Percolation in the signal to interference ratio graph

2006 ◽  
Vol 43 (02) ◽  
pp. 552-562 ◽  
Author(s):  
Olivier Dousse ◽  
Massimo Franceschetti ◽  
Nicolas Macris ◽  
Ronald Meester ◽  
Patrick Thiran
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Euclidean Distance ◽  
Critical Density ◽  
Continuum Percolation ◽  
Two Dimensional ◽  
Radio Communications ◽  
Communications Networks ◽  
Infinite Range ◽  
Independent Model

Continuum percolation models in which pairs of points of a two-dimensional Poisson point process are connected if they are within some range of each other have been extensively studied. This paper considers a variation in which a connection between two points depends not only on their Euclidean distance, but also on the positions of all other points of the point process. This model has been recently proposed to model interference in radio communications networks. Our main result shows that, despite the infinite-range dependencies, percolation occurs in the model when the density λ of the Poisson point process is greater than the critical density value λc of the independent model, provided that interference from other nodes can be sufficiently reduced (without vanishing).

Poisson approximation related to spectra of hierarchical Laplacians

2019 ◽  
Vol 20 (05) ◽  
pp. 2050035
Author(s):  
Alexander Bendikov ◽  
Wojciech Cygan
Keyword(s):  
Continuous Function ◽  
Point Process ◽  
Density Of States ◽  
Poisson Point Process ◽  
Point Processes ◽  
Point Spectrum ◽  
Random Variables ◽  
Poisson Approximation ◽  
Ultrametric Space ◽  
Locally Compact

Let [Formula: see text] be a locally compact separable ultrametric space. Given a measure [Formula: see text] on [Formula: see text] and a function [Formula: see text] defined on the set of all non-singleton balls [Formula: see text] of [Formula: see text], we consider the hierarchical Laplacian [Formula: see text]. The operator [Formula: see text] acts in [Formula: see text] is essentially self-adjoint and has a purely point spectrum. Choosing a sequence [Formula: see text] of i.i.d. random variables, we consider the perturbed function [Formula: see text] and the perturbed hierarchical Laplacian [Formula: see text] Under certain conditions, the density of states [Formula: see text] exists and it is a continuous function. We choose a point [Formula: see text] such that [Formula: see text] and build a sequence of point processes defined by the eigenvalues of [Formula: see text] located in the vicinity of [Formula: see text]. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity [Formula: see text].

scholarly journals Effect of APOE ε4 on multimodal brain connectomic traits: a persistent homology study

2020 ◽  
Vol 21 (S21) ◽  
Author(s):  
Jin Li ◽  
◽  
Chenyuan Bian ◽  
Dandan Chen ◽  
Xianglian Meng ◽  
...  
Keyword(s):  
Magnetic Resonance Imaging ◽  
Magnetic Resonance ◽  
Genetic Risk ◽  
Brain Connectivity ◽  
Persistent Homology ◽  
Brain Network ◽  
Resonance Imaging ◽  
Modeling Framework ◽  
Apoe Ε4 ◽  
Topological Features

Abstract Background Although genetic risk factors and network-level neuroimaging abnormalities have shown effects on cognitive performance and brain atrophy in Alzheimer’s disease (AD), little is understood about how apolipoprotein E (APOE) ε4 allele, the best-known genetic risk for AD, affect brain connectivity before the onset of symptomatic AD. This study aims to investigate APOE ε4 effects on brain connectivity from the perspective of multimodal connectome. Results Here, we propose a novel multimodal brain network modeling framework and a network quantification method based on persistent homology for identifying APOE ε4-related network differences. Specifically, we employ sparse representation to integrate multimodal brain network information derived from both the resting state functional magnetic resonance imaging (rs-fMRI) data and the diffusion-weighted magnetic resonance imaging (dw-MRI) data. Moreover, persistent homology is proposed to avoid the ad hoc selection of a specific regularization parameter and to capture valuable brain connectivity patterns from the topological perspective. The experimental results demonstrate that our method outperforms the competing methods, and reasonably yields connectomic patterns specific to APOE ε4 carriers and non-carriers. Conclusions We have proposed a multimodal framework that integrates structural and functional connectivity information for constructing a fused brain network with greater discriminative power. Using persistent homology to extract topological features from the fused brain network, our method can effectively identify APOE ε4-related brain connectomic biomarkers.

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